Literature review (iv)

This is the fourth part of the literature review, as submitted in part for the annual review and which will eventually form the introduction to my thesis. For more details on this series, please see the first part.

The classification of symmetries, linearising transformations and invariant soltuions formed a great deal of the research effort once symmetry methods were reclaimed from pure mathematicians. Ovsiannikov paid much attention to constructing group-invariant solutions, whilst a reference book containing the symmetries etc. of many PDEs was developed by Ibragimov.

Thus, in its classical form, Lie's approach has a vast array of applications to differential equations and their solutions. However, there are many extensions to the classical symmetry method that further the uses of symmetry analysis as a whole.

Lie's approach of utilising point symmetries — which act on the space of independent and dependent variables — can be extended to transformations that act upon derivatives of both dependent and independent variables. For example, symmetries that depend upon first-order derivatives of the dependent variables are known as contact symmetries; hence, every Lie point symmetry is a Lie contact symmetry. Whereas the associated characteristic of point symmetries is linear in y', this is not necessarily the case for contact symmetries. Still, point and contact symmetries are both geometrical transformations defined independently of any specific ODE: they are respectively diffeomorphisms of the plane and J^{1} and can both be extended to higher Jet spaces by a process known as prolongation. Although there are no other transformations that can be prolonged in such a way, that does not mean that such transformations are not useful. Indeed, we only need consider those transformations that act as diffeomorphisms on the subset of the Jet space defined by the differential equation in question. Such transformations are known as "generalised" symmetries and are those Lie groups whose infinitesimal generators depend on derivatives of the dependent and independent variables up to a finite order. If they depend only on first-order derivatives they are thus contact symmtries. Some inconsistency in the literature results in many different names for generalised symmetries: they are sometimes referred to as dynamical or internal symmetries, sometimes as Lie-Backlund transformations and also "higher-order" symmetries. Olver contains an interesting discussion regarding the curious history of generalised symmetries.

The concept of a generalised symmetry was introduced by Emmy Noether in her celebrated 1918 paper in order to show how variational symmetries of differential equations lead to conservation laws for the corresponding Euler-Lagrange equations. Her famous theorem proved that a generalised symmetry admitted by the variational complex of the associated E-L equation gives rise to a conservation law for the governing differential equations. For example, the translational invariance of time gives rise to the conservation of energy and rotational invariance determines the conservation of angular momentum. Noether's theorem provided the framework in which mathematicians could therefore complete a systematic investigation of the symmetry properties and conservation laws of various important mathematical problems, a direct and fundamental consequence of the work of Lie and his development of symmetry methods.